There is no obvious way to integrate the secant function, sec(x). One way is to use the method of partial fractions. While this method takes a quite a few steps to solve the integral, it is robust. It does not rely heavily in the intuition of the student.

Another way involves some trickery where we try to form an integral of the form f'(x)/f(x), which gives the result ln|f(x)|. Although this method is efficient, it requires you to know part of the answer before you begin, which doesn't make much sense.

Using the method of partial fractions, the idea is to rewrite the integral of sec(x) as:

∫sec(x)dx = ∫cos(x)dx / (1 - sin^2(x))

Then by letting u = sin(x)...

∫sec(x)dx = ∫du / (1 - u^2) = ∫ du / (1+u)(1-u)

We can then convert the expression 1/(1+u)(1-u) into the sum of its partial fractions.

Suggested video:

Partial Fraction Decomposition: [ Ссылка ]

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